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Abstract
A characteristic matrix function captures the spectral information of a bounded linear operator in a matrix-valued function. In this article, we consider a delay differential equation with one discrete time delay and assume this equation is equivariant with respect to a compact symmetry group. Under this assumption, the delay differential equation can have discrete wave solutions, i.e. periodic solutions that have a discrete group of spatio-temporal symmetries. We show that if a discrete wave solution has a period that is rationally related to the time delay, then we can determine its stability using a characteristic matrix function. The proof relies on equivariant Floquet theory and results by Kaashoek and Verduyn Lunel on characteristic matrix functions for classes of compact operators. We discuss applications of our result in the context of delayed feedback stabilization of periodic orbits.
1. Introduction
For infinite dimensional dynamical systems, determining the stability of an invariant set often poses challenges due to the infinite dimensional nature of the problem. However, sometimes we are able to make a dimension reduction in the sense that the stability of the invariant set can be determined by computing zeros of a scalar valued function. This simplifies the stability analysis since we can now apply analytical and numerical techniques directly to the scalar valued function. Concrete examples of this approach appear in the context of partial differential equations, where the stability of certain travelling wave solutions can be computed using the Evans function [Citation20]. For delay differential equations (DDE), the stability of equilibria and certain classes of periodic orbits can be computed using so-called characteristic matrix functions, as introduced by Kaashoek and Verduyn Lunel in [Citation15] and [Citation16].
This article is concerned with DDE that have built-in symmetries; in this case, periodic solutions of the DDE can satisfy additional spatio-temporal relations. Although spatio-temporal patterns and their stability are well studied in the context of ordinary differential equations (see, e.g. [Citation5, Citation17, Citation26]), they are much less explored in the setting of DDE. This article makes a next step in the stability analysis of spatio-temporal patterns in DDE by combining the concept of characteristic matrix functions with techniques from symmetric systems.
Specifically, we consider a DDE of the form
(1)
(1) with
a
function and time delay
. We assume that the DDE (Equation1
(1)
(1) ) is equivariant with respect to a compact subgroup
of the general linear group. In this case, a periodic orbit
of (Equation1
(1)
(1) ) can satisfy spatio-temporal relations of the form
, with
a spatial transformation and r>0 a fraction of the period. We prove that if the time shift r is equal to the time delay τ, we can determine the stability of
by computing the zeros of a scalar valued function.
In the situation where the DDE (Equation1(1)
(1) ) is not symmetric but does have a periodic orbit with period equal to the delay, we can determine the stability of this periodic orbit by computing the zeros of a scalar valued function. This result was proven under an additional condition in [Citation25] and later in full generality in [Citation16, Section 11.4]. The result presented in this article can be viewed as a refinement of the result in [Citation25] and [Citation16, Section 11.4] in the sense that under the extra assumption that the DDE is symmetric, we are able to make more precise statements about periodic solutions with spatio-temporal patterns.
The result presented in this article is particularly relevant in the context of equivariant Pyragas control, a delayed feedback control scheme that aims to stabilize spatio-temporal patterns. If the ordinary differential equation
(2)
(2) has an unstable periodic solution
with spatio-temporal relation
, then this is also a solution of the delay differential equation
(3)
(3) cf. [Citation9]. However, the overall dynamics of systems (Equation2
(2)
(2) ) and (Equation3
(3)
(3) ) are radically different, and we can try to choose the matrix
in such a way that
is a stable solution of (Equation3
(3)
(3) ). Despite its many experimental applications (see, e.g. [Citation4, Citation11]), mathematical results on the control scheme (Equation3
(3)
(3) ) are rare due to the periodic and infinite dimensional nature of the problem. In fact, most analytical results in the literature so far are either close to a bifurcation point [Citation2, Citation7, Citation10, Citation12, Citation14] or concern periodic orbits that can be transformed to stationary solutions of autonomous systems [Citation6, Citation8, Citation9, Citation18, Citation21]. The contents of this article are a first step towards further insights in equivariant Pyragas control, such as the results on stabilization of non-stationary periodic orbits far away from bifurcation point presented by the author in [Citation1].
We start the rest of this article by formally stating its main result and introducing the necessary terminology in Section 2. Section 3 then reviews material from [Citation15] and [Citation16] on characteristic matrix functions. Section 4 contains the proof of the main result. In Section 5, we further discuss applications of the result to delayed feedback control of periodic orbits.
2. Setting and statement of the main result
Throughout this section, we consider the DDE (Equation1(1)
(1) ) and assume that this DDE has a periodic orbit and is symmetric with respect to a compact subgroup of the general linear group. We summarize this in the following hypothesis:
Hypothesis 2.1
DDE (Equation1
(1)
(1) ) has a periodic orbit
with minimal period p>0;
DDE (Equation1
(1)
(1) ) is equivariant with respect to a compact subgroup Γ of the general linear group
, i.e.
(4)
(4)
If is a solution of the DDE (Equation1
(1)
(1) ) and γ is an element of Γ, then the equivariance relation (Equation4
(4)
(4) ) implies that
is a solution of the DDE (Equation1
(1)
(1) ) as well. So we can view Γ as a group of symmetries of the solutions of (Equation1
(1)
(1) ). Moreover, the equivariance relation (Equation4
(4)
(4) ) naturally induces two symmetry groups on the periodic orbit
; we define the groups:
(5a)
(5a)
(5b)
(5b) The elements of group
leave the periodic orbit fixed pointwise; therefore we refer to
as the group of spatial symmetries of
. Since every element
induces a spatio-temporal relation of the form
on the periodic solution, we refer to
as the group of spatio-temporal symmetries of
.
If are two spatio-temporal symmetries of
, then
and hence
. Thus the map
is a group homomorphism and
is a normal subgroup of
. Therefore
and
is a subgroup of
. This implies that
where
denotes the cyclic group of order m. If
, we say that the periodic solution
is a rotating wave; if
(i.e. the group of spatio-temporal symmetries modulo the group of purely spatial ones is a finite group), we say that the periodic solution
is a discrete wave, cf. [Citation5].
To determine whether is a stable solution of the DDE (Equation1
(1)
(1) ), we consider the linearized system
(6a)
(6a) If we supplement (Equation6a
(6a)
(6a) ) with the initial condition
(6b)
(6b) then the system (Equation6a
(6a)
(6a) )–(Equation6b
(6b)
(6b) ) has a unique solution
for
. We define the history segment
of this solution as
. We then associate to (Equation6a
(6a)
(6a) )–(Equation6b
(6b)
(6b) ) a two-parameter system of operators
(7)
(7) defined via the relation
. We refer to (Equation7
(7)
(7) ) as the family of solution operators of (Equation6a
(6a)
(6a) ), cf. [Citation3, Chapter 12].
Floquet theory for DDE implies that the non-zero spectrum of the monodromy operator consists of isolated eigenvalues of finite algebraic multiplicity; these eigenvalues of
determine the stability of the periodic solution
[Citation3, Chapter 13]. The equivariance assumption in Hypothesis 2.1 allows us to refine Floquet theory for discrete waves. We make this precise in the following proposition, which we cite without proof from [Citation1]. The statement of the proposition is analogous to the formulation of equivariant Floquet theory for ODE (cf. [Citation26]), but the proof now also involves the compactness of the relevant operator.
Throughout, we let an element act on the state space
via
for
and
.
Proposition 2.1
Stability of discrete waves, [Citation1, Proposition 6.3]
Consider the DDE (Equation1(1)
(1) ) satisfying Hypothesis 2.1 and additionally assume that the periodic solution
is a discrete wave. Let
be the family of solution operators of the linearized problem (Equation6a
(6a)
(6a) ). For
a spatio-temporal symmetry of
, definethe operator
(8)
(8) Then the following statements hold:
The non-zero spectrum of
consists of isolated eigenvalues of finite algebraic multiplicity.
is an eigenvalue of
.
If
has an eigenvalue strictly outside the unit circle, then
is an unstable solution of (Equation1
(1)
(1) ). If the eigenvalue
is algebraically simple and all other eigenvalues of
lie strictly inside the unit circle, then
is a stable solution of (Equation1
(1)
(1) ).
We aim to determine the eigenvalues of the operator in (Equation8
(8)
(8) ) using the concept of a characteristic matrix function for bounded linear operators, as introduced by Kaashoek and Verduyn Lunel in [Citation16]. We denote by
the space of bounded linear operators on a complex Banach space X. Moreover, we denote by
the identity operator on a Banach space X, but suppress the subscript when the underlying space is clear.
Definition 2.2
[Citation16, Definition 5.2.1]
Let X be a complex Banach space, be a bounded linear operator and
be an analytic matrix-valued function. We say that Δ is a characteristic matrix function for T if there exist analytic functions
such that
are invertible operators for all
and such that
(9)
(9) holds for all
.
The idea of the above definition is to make a conjugation between the analytic function
and the analytic function
However, this cannot be done directly, since in general the dimensions of X and
are not the same. So we first (trivially) extend the functions
and
to the functions
(10)
(10) respectively. If now the functions in (Equation10
(10)
(10) ) are related via multiplication by analytic functions whose values are invertible operators, then this directly relates kernelvectors of I−zT to kernelvectors of
. In particular, zeros of the scalar valued function
give information on the non-zero spectrum of T. We make this precise in Section 3. We first state the main result of this article, which gives an explicit characteristic matrix functionfor the operator
defined in (Equation8
(8)
(8) ) in case the time shift
of the spatio-temporal pattern is equal to the time delay τ.
Theorem 2.3
Main result
Consider the DDE
(11)
(11) with
a
-function and with time delay
. Assume that
system (Equation11
(11)
(11) ) has a periodic solution
with minimal period p>0;
system (Equation11
(11)
(11) ) is equivariant with respect to a compact subgroup Γ of the general linear group
;
the periodic solution
is a discrete wave and there exists a spatio-temporal symmetry
with
3. Characteristic matrix functions and spectral information
This section reviews material from [Citation15, Section 1] and [Citation16, Chapter 5] on characteristic matrix functions. The first part of this section (Definition 3.1-Lemma 3.4) discusses how characteristic matrix functions capture the spectrum of a bounded linear operator. We have included the contents of Definition 3.1-Lemma 3.4 in this article to give context to Theorem 2.3 and to illustrate the implications of this theorem; we discuss its applications further in Section 5. In the second part of this section (Theorem 3.5), we state a theorem from [Citation16] that constructs a characteristic matrix function for a class of compact operators. This theorem is the cornerstone for the proof of Theorem 2.3 and is therefore crucial for the rest of this article.
We start by recalling the notion of Jordan chains for analytic operator-valued functions.
Definition 3.1
Let X be a complex Banach space and an analytic operator-valued function. Given a complex number
, we say that an ordered set
of vectors in X is a Jordan chain of lengthk for L at μ if
and
(13)
(13) The maximal length of a Jordan chain starting with
is called the rank of
; the rank is said to be infinite if no maximum exists.
Example 3.2
cf. [Citation15, p. 485]
Given a bounded linear operator , the usual notion of a Jordan chain for T coincides with the notion of a Jordan chain for the analytic function
Indeed, let
be an eigenvalue of T and let
be an associated Jordan chain, i.e.
(14)
(14) Then
So
(15)
(15) and
is a Jordan chain for the analytic function
. Vice versa, suppose the vectors
satisfy (Equation15
(15)
(15) ). Then evaluating the derivatives of (Equation15
(15)
(15) ) at
yields the equalities (Equation14
(14)
(14) ) and thus
is a Jordan chain for the bounded operator T.
If is Jordan chain for L at μ, then (Equation13
(13)
(13) ) implies that
. Vice versa, if
satisfies
, then
and hence L has a Jordan chain (of at least length 1) at μ starting with
. So L has a Jordan chain at μ starting with
if and only
is a non-zero element of the space
We now consider the case in which the space
is finite dimensional and all Jordan chains of L at μ have finite rank. We pick a basis
of
and for
, we let
be the rank of
. Then, if
is an element of
, its rank has to be equal to 1 of the
. In particular, the set
does not depend on the choice of basis. We define the algebraic multiplicity of μ as the number
The next lemma shows that the algebraic multiplicity is invariant under conjugation with analytic matrix-valued functions whose values are invertible operators.
Lemma 3.3
[Citation15, Proposition 1.2], [Citation16, Proposition 5.1.1]
Given a complex Banach space X, let
be analytic operator-valued functions, and let
be analytic operator-valued functions whose values are invertible operators. Suppose that
for all
. Then, for
, the algebraic multiplicity of L at μ equals the algebraic multiplicity of M at μ.
Proof.
We show that there is a one-to-one correspondence between Jordan chains for L at μ and Jordan chains for M at μ; from there the claim follows.
Let be a Jordan chain for L at μ, i.e.
For
, let
be such that
Then
and
so
satisfy
So
is a Jordan chain for M at μ. Vice versa, every Jordan chain
of M at μ induces a Jordan chain for L at μ. So there is a one-to-one correspondence between Jordan chains for M at μ and Jordan chains for L at μ. In particular, the algebraic multiplicity of L at μ equals the algebraic multiplicity of M at μ.
We are now ready to make precise how a characteristic matrix function, as defined in Definition 2.2, captures the spectral information of a bounded linear operator:
Lemma 3.4
[Citation16, Theorem 5.2.2]
Let X be a complex Banach space, a bounded linear operator and
a characteristic matrix function for T. Let
, then
if and only if
, i.e.
If
, then the geometric multiplicity of
as an eigenvalue of T equals the dimension of the space
If
, then the algebraic multiplicity of
as an eigenvalue of T equals the order of μ as a root of
(16)
(16)
Proof.
Let , then
is an eigenvalue of T if and only if
has a non-trivial kernel, i.e. if and only if
has a non-trivial kernel. We first show that there is a one-to-one correspondence between kernel vectors of
and kernel vectors of
. This then implies the first two statements of the lemma.
We write
(17)
(17) then the kernels of the operators
are given by
Since Δ is a characteristic matrix function for T, there exists analytic functions
so that
are invertible operators for all
and such that
for all
. In particular, the operator
maps the space
in a one-to-one way to the space
. This implies that the map
with inverse
is a bijection. So there is a one-to-one correspondence between elements of
and elements of
, which proves the first two statements of the lemma.
To prove the third statement of the lemma, we first show that for any number , the algebraic multiplicity of Δ at μ equals the order of μ as a root of the equation
. To do so, we bring Δ in local Smith form: given
, there exists analytic functions G, H, whose values are invertible matrices, and unique non-negative integers
such that
(18a)
(18a) with
(18b)
(18b) see [Citation15, Section 1]. The algebraic multiplicity of D at μ equals
; therefore, Lemma 3.3 implies that the algebraic multiplicity of Δ at μ equals
as well. On the other hand, by (Equation18a
(18a)
(18a) )–(Equation18b
(18b)
(18b) ) we can write
as
Since
are invertible matrices, the order of μ as a root of
is also given by
. We conclude that the algebraic multiplicity of Δ at μ equals the order of μ as a root of
.
Now let . Then by equality (Equation9
(9)
(9) ) and Lemma 3.3, the algebraic multiplicity of
as an eigenvalue of T equals the algebraic multiplicity of Δ at μ; by the previous step, this equals the order of μ as a root of
. We conclude that the algebraic multiplicity of
as an eigenvalue of T equals the order of μ as a root of
, as claimed.
The next theorem from [Citation16] gives a sufficient condition for a bounded linear operator to have a characteristic matrix function. The proof of this theorem is beyond the scope of this article and hence we state the theorem without proof.
Theorem 3.5
[Citation16, Theorem 6.1.1]
Let X be a complex Banach space and a bounded linear operator. Assume that T is of the form T = V + R with
a Volterra operator, i.e. V is compact and
;
R an operator of finite rank
.
Decompose R as R = DC where
Then the matrix-valued function
is a characteristic matrix function for T.
Note that, since V is Volterra, the resolvent map is analytic on
and hence
is an analytic matrix-valued function.
4. Characteristic matrix functions for DDE with symmetries
In this section, we prove this article's main result Theorem 2.3. We prove this theorem by writing the operator (Equation8(8)
(8) ) as the sum of a Volterra operator and a finite rank operator; we then apply Theorem 3.5 to obtain a characteristic matrix function.
In [Citation16, Section 11.4], the authors consider a DDE with a periodic solution; they do not assume any symmetry relations on the DDE, but do assume that the period of the periodic solution is equal to the time delay. In this setting, they construct a characteristic matrix function for the monodromy operator. The difference between Theorem 2.3 presented here and the result in [Citation16, Section 11.4] is the following: we realize that in a symmetric setting, the operator (Equation8(8)
(8) ) has a characteristic matrix function if the time shift of the spatio-temporal symmetry is equal to the time delay, whereas in the non-symmetric case considered in [Citation16, Section 11.4] the monodromy operator has a characteristic matrix function if the period of the periodic solution is equal to the time delay. The proof of Theorem 2.3 is similar in spirit to the arguments in [Citation16, Section 11.4], but we additionally exploit the equivariance of the considered system.
This section is structured as follows: in Section 4.1 we first consider a linear, time-dependent DDE whose coefficients satisfy a spatio-temporal relation; for this DDE we construct a characteristic matrix function. The proof of Theorem 2.3 then follows in Section 4.2.
4.1. Linear, time-dependent DDE with spatio-temporal symmetry
We consider the initial value problem
(19a)
(19a) with time delay
and initial condition
at time
. We make the following assumptions on system (Equation19a
(19a)
(19a) ):
Hypothesis 4.1
the functions
are
;
there exists an invertible matrix
such that
(19b)
(19b) for all
.
We stress that the time shift τ in Equation (Equation19b(19b)
(19b) ) is the same as the time delay of the DDE (Equation19a
(19a)
(19a) ). So the coefficients A, B satisfy some spatio-temporal relation with time shift equal to the time delay of (Equation19a
(19a)
(19a) ). Under this hypothesis, we construct a characteristic matrix function Δ for the operator
, where
is the family of solution operators of (Equation19a
(19a)
(19a) ). We give an explicit expression for Δ in terms of solutions of the family of ODE
(20)
(20) with
. To arrive at this expression for Δ, we make the following intermediate steps:
We show that the symmetry relations (Equation19b
(19b)
(19b) ) imply symmetry relations on the fundamental solution of the ODE (Equation20
(20)
(20) ) (Lemma 4.1).
We give an explicit expression for
(Lemma 4.2) and write
, with V an integral operator and R an operator of finite rank.
We show that the integral operator V is in fact a Volterra operator (Lemma 4.3).
We apply Theorem 3.5 to find a characteristic matrix functionfor
(Proposition 4.4).
In the case where the coefficients A, B of (Equation20(20)
(20) ) are periodic, i.e. when
for some p>0, the fundamental solution
of (Equation20
(20)
(20) ) satisfies the additional relation
(21)
(21) Indeed, the matrix-valued function
satisfies the ODE
with initial condition
for t = s. So the uniqueness of solutions implies (Equation21
(21)
(21) ). Similarly, the symmetry relation (Equation19b
(19b)
(19b) ) on the coefficients A, B induces symmetry relations on the fundamental solution
, as we make precise in the following lemma:
Lemma 4.1
Consider functions satisfying Hypothesis 4.1. For
, let
be the fundamental solution of the ODE
with
. Then it holds that
(22)
(22) for all
.
In particular, if is the fundamental solution of the ODE
with
, then
(23)
(23)
Proof.
The matrix-valued function satisfies the ODE
with initial condition
for
. Similarly, the matrix-valued function
satisfies the ODE
with initial condition
for
. The uniqueness of solutions now implies the relation (Equation22
(22)
(22) ).
Since when
, the equality (Equation22
(22)
(22) ) implies the equality (Equation23
(23)
(23) ).
We next give an explicit expression for the operator .
Lemma 4.2
Consider the DDE (Equation19a(19a)
(19a) ) satisfying Hypothesis 4.1 and let
be the family of solution operators of (Equation19a
(19a)
(19a) ). Moreover, let
be the fundamental solution of the ODE
with
. Then the operator
is given by
(24)
(24)
Proof.
For s = 0 and , the initial value problem (Equation19a
(19a)
(19a) ) becomes
which we solve by the Variation of Constants formula as
(25)
(25) With
, (Equation25
(25)
(25) ) becomes
where in the last step we used (Equation19b
(19b)
(19b) ) and (Equation23
(23)
(23) ). So
is given by
which proves the lemma.
To apply Theorem 3.5, we first complexify the operator in (Equation24
(24)
(24) ) via a canonical procedure as detailed in, for example, [Citation3, Chapter 3.7]. However, we do not make the complexification explicit in notation, i.e. we write
both for the real operator on the real Banach space
and the complexified operator on the complex Banach space
. We then decompose the complex operator
as
(26)
(26) with the (suggestive) notation
(27a)
(27a)
(27b)
(27b) and complex Banach space X given by
We next prove that the integral operator (Equation27a
(27a)
(27a) ) is in fact a Volterra operator.
Lemma 4.3
The operator V defined in (Equation27a(27a)
(27a) ) is Volterra, i.e. V is compact and
.
Proof.
We first prove that . To do so, fix
and let
be such that
, i.e.
(28)
(28) Equality (Equation28
(28)
(28) ) implies that
. Moreover, since the left-hand side of (Equation28
(28)
(28) ) is
, the right-hand side is
as well; differentiating both sides with respect to θ gives
(29)
(29) Equality (Equation28
(28)
(28) ) also implies that
and substituting this into (Equation29
(29)
(29) ) gives
So ϕ satisfies the initial value problem
which implies that
. We conclude that
is not an eigenvalue of V, and thus that
.
If , then (Equation27a
(27a)
(27a) ) implies that
and hence by the Arzelà–Ascoli theorem V is compact. This implies that the non-zero spectrum of V consists of eigenvalues. Since we already showed that
, we conclude that
. So V is a compact operator and
, which proves the claim.
We are now ready to use Theorem 3.5 and give an explicit characteristic matrix function for the operator :
Proposition 4.4
Consider the DDE (Equation19a(19a)
(19a) ) satisfying Hypothesis 4.1; let
be the family of solution operators of (Equation19a
(19a)
(19a) ). Moreover, for
, let
be the fundamental solution of the ODE
with
. Then the matrix-valued function
(30)
(30) is a characteristic matrix functionfor the operator
(31)
(31)
Proof.
We divide the proof into two steps:
Step 1: The finite rank operator R in (27b) factorizes as R = DC with
(32a)
(32a)
(32b)
(32b) For V as in (Equation27a
(27a)
(27a) ) and D as in (Equation32b
(32b)
(32b) ), we now give an explicit expression for
. To that end, fix
and
; let
be the unique element such that
, i.e. ϕ satisfies
(33)
(33) Equality (Equation33
(33)
(33) ) implies that
. Moreover, since the right-hand side of (Equation33
(33)
(33) ) is
, the left-hand side is
as well; differentiating both sides with respect to θ gives
(34)
(34)
(35)
(35) where in the last step we used (Equation19b
(19b)
(19b) ). Equality (Equation33
(33)
(33) ) also implies that
and substituting this into (Equation35
(35)
(35) ) gives that
So ϕ satisfies the initial value problem
which implies that
. Equality (Equation22
(22)
(22) ) with
and
implies that
and hence
So we conclude that
(36)
(36) Step 2: We now prove the statement of the proposition. The operator
decomposes as
with V defined in (Equation27a
(27a)
(27a) ) and R defined in (27b). The operator R is a finite rank operator; by Lemma 4.3, the operator V is a Volterra operator. Therefore, if we let D, C be as in (Equation32a
(32a)
(32a) )–(Equation32b
(32b)
(32b) ), Theorem 3.5 implies that
is a characteristic matrix function for
. Equality (Equation36
(36)
(36) ) implies that
and hence
is a characteristic matrix function for
, as claimed.
4.2. Proof of Theorem 2.3
Theorem 2.3 now follows from Proposition 4.4:
Proof
Proof of Theorem 2.3
Define
We show that the coefficients A, B satisfy Hypothesis 4.1. By assumption, the periodic solution
has a spatio-temporal symmetry
with
Moreover, since
satisfies the equivariance relation (Equation4
(4)
(4) ), it holds that
for all
and i = 1, 2. So it in particular holds that
and similarly
So the coefficients A, B satisfy Hypothesis 4.1; therefore Proposition 4.4 implies Theorem 2.3.
We considered the system (Equation19a(19a)
(19a) )–(Equation19b
(19b)
(19b) ) with in the back of our mind the linearized DDE (Equation12
(12)
(12) ). However, Equations (Equation19a
(19a)
(19a) )–(Equation19b
(19b)
(19b) ) also cover the special case h = I. In this case, Equation (Equation19a
(19a)
(19a) ) has periodic coefficients with period equal to the time delay, and the operator (Equation31
(31)
(31) ) is the monodromy operator. So in this case, an application Proposition 4.4 gives a characteristic matrix function for the monodromy operator, and we recover the result from [Citation16, Section 11.4]:
Theorem 4.5
cf. [Citation16, Section 11.4]
Consider the DDE
(37)
(37) with
a
function and with time delay
. Assume that system (Equation37
(37)
(37) ) has a periodic solution
with period τ, i.e.
Let
be the family of solution operators associated to the linearized DDE
For
, let
be the fundamental solution of the ODE
with
. Then the analytic function
is a characteristic matrix function for the monodromy operator
Proof.
Define
then it holds that
So the coefficients A, B satisfy Hypothesis 4.1 with
. Therefore Proposition 4.4 implies the statement of the theorem.
5. Applications to delayed feedback control
In [Citation19], Pyragas introduced a delayed feedback method (now known as Pyragas control) that aims to stabilize periodic orbits of the ordinary differential equation
(38)
(38) The feedback term introduced by Pyragas measures the difference between the current state and the state time τ ago, and feeds this difference (multiplied by a matrix) back into the system. Concretely the system with feedback control becomes
(39)
(39) with time delay
and matrix
. If now
is a τ-periodic solution of (Equation38
(38)
(38) ), then it is also a solution of (Equation39
(39)
(39) ). However, the overall dynamics of the systems with and without feedback are different, and it is possible that
is an unstable solution of (Equation38
(38)
(38) ) but a stable solution of (Equation39
(39)
(39) ).
We can determine whether is a stable solution of (Equation39
(39)
(39) ) by computing the eigenvalues of the monodromy operator
where
is the family of solution operators of the DDE
The results in [Citation15], [Citation16, Section 11.4] (cf. Theorem 4.5 in this article) give a characteristic matrix function of the monodromy operator
in terms of solutions of the ODE
The explicit expression for a characteristic matrix function for
has for example been used in the context of feedback control of a Hamiltonian system [Citation10] and in studying the behaviour of the control scheme (Equation39
(39)
(39) ) as the delay τ goes to infinity [Citation23].
Equivariant Pyragas control [Citation9] adapts the Pyragas feedback scheme so that the feedback term vanishes on a periodic orbit with a specific spatio-temporal pattern. More precisely, suppose that
(Equation38
(38)
(38) ) is equivariant with respect to a compact symmetry group
;
(Equation38
(38)
(38) ) has a periodic solution
with minimal period p>0;
is a discrete wave and
is a spatio-temporal symmetry of
, i.e.
for some
.
Then the periodic solution is also a solution of the feedback system
(40)
(40) with
. We additionally make the mild assumption that the matrix
satisfies hK = Kh, so that the system (Equation40
(40)
(40) ) is again equivariant with respect to the group generated by h.
In system (Equation40(40)
(40) ), the delay
is strictly smaller than the minimal period of
, and hence we are not in the setting of Theorem 4.5. However, Theorem 2.3 gives a characteristic matrix function
for the operator
where
is the family of solution operators of the DDE
(41)
(41) The eigenvalues of the operator
determine whether
is stable as a solution of (Equation40
(40)
(40) ) (cf. Proposition 2.1); therefore, we can establish whether the control scheme (Equation40
(40)
(40) ) succeeds or fails to stabilize
by computing the roots of the equation
(see also Lemma 3.4). This result contributes to the current literature on equivariant Pyragas control in two ways:
To prove that
is an unstable solution of (Equation40
(40)
(40) ), it suffices to find (at least) one eigenvalue of the operator
outside the unit circle; and in specific situations, it is indeed possible to do exactly that [Citation13]. However, if we want to establish that
is a stable solution of (Equation40
(40)
(40) ), we have to ensure that we find all non-zero eigenvalues of
and have to be careful about the multiplicity of the trivial eigenvalue
. Theorem 2.3 paves a way to do that, since the characteristic matrix function captures all non-zero eigenvalues of
and also captures both their geometric and their algebraic multiplicity (cf. Lemma 3.4).
In the literature so far, most analytical results on successful equivariant Pyragas control are either close to a bifurcation point [Citation2, Citation7, Citation10, Citation12, Citation14] or consider rotating waves, i.e. periodic solutions that can be transformed to stationary states of autonomous systems [Citation6, Citation8, Citation9, Citation18, Citation21]. Both these approaches simplify the stability analysis, but also work only in specific settings, i.e. they strongly depend on the form of the ODE (Equation38
(38)
(38) ). In the context of equivariant Pyragas control, Theorem 2.3 also simplifies the stability analysis by reducing the infinite dimensional problem to a finite dimensional one. However, this simplification is general in the sense that it does not depend on the specific form of the ODE (Equation38
(38)
(38) ). Therefore, we believe that Theorem 2.3 is a first step in proving new stabilization results (such as the stabilization results for non-stationary periodic solutions and far away from bifurcation point in [Citation1]) and will generally be a helpful tool in further developments in equivariant Pyragas control.
6. Discussion
In [Citation24], Szalai et al. discuss a delay equation of the form
that has a periodic solution of period
. To find geometrically simple eigenvalues of this periodic orbit, they construct a characteristic matrix function that takes values in
. In general, if the delay equation
has a periodic orbit with period
, one expects that monodromy operator has a characteristic matrix function taking values in
, see also [Citation22]. In Theorem 2.3, in contrast, the period of the periodic orbit of (Equation11
(11)
(11) ) is rationally related to the delay, but the constructed characteristic matrix function takes values in
. The difference here is that we do not construct a characteristic matrix function for the monodromy operator, but exploit the equivariance relations and construct a characteristic matrix function for the operator (Equation8
(8)
(8) ). So working with the operator (Equation8
(8)
(8) ) also has a computational advantage, since it yields a lower dimensional characteristic matrix function.
Throughout this article, we studied the stability of periodic orbits of DDE using the principle of linearized stability, i.e. by studying the behaviour of the linearized system. The advantage of this is that for linear DDE of the form
one can very explicitly compute the time τ-map, cf. Lemma 4.2. In contrast, a Poincaré map for periodic orbits of DDE can be constructed abstractly [Citation3, Section 14.3], but in general no explicit expression for the Poincaré map is available.
Acknowledgements
The author is grateful to Bernold Fiedler and Sjoerd Verduyn Lunel for useful discussions and encouragement; and to Alejandro López Nieto, Bob Rink, Isabelle Schneider and an anonymous referee for comments on earlier versions. The contents of this article are based upon contents of the authors doctoral thesis [Citation1], written at the Freie Universität Berlin under the supervision of Bernold Fiedler.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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